Mysteries and marvels of rational base numeration systems

The definition of numeration systems with rational base, in a joint work
with S.~Akiyama and Ch.~Frougny (Israel J. Math. 2008), has allowed to
make some progress in a number theoretic problem, by means of automata
theory and combinatorics of words.  At the same time, it raised the
problem of understanding the structure of the sets of the
representations of the integers in these systems from the point of view
of formal language theory.

At first sight, these sets look rather chaotic and do not fit well
in the classical Chomsky hierarchy of languages. They all enjoy a
property that makes them defeat, so to speak, any kind of iteration
lemma. On the other hand, these sets also exhibit remarkable
regularity properties.

During the recent years, these regularities have been studied in a
series of joint papers with my student V. Marsault. In particular, we
have shown that periodic signatures are characteristic of the
representation languages in rational base numeration systems and
studied, jointly with S. Akiyama, a kind of autosimilarity property that
also leads to the construction of Cantor-like sets.

The representation languages still keep most of their mystery. The
partial results which will be presented call for further investigations
on the subject even stronger.